Toward a Temporal Scale for Emergent Spacetime: Spatial SU(2) Invariance Leaves the Temporal Coefficient Free

Current Zenodo release: version 2.0 (2026-09-28). Official title: Toward a Temporal Scale for Emergent Spacetime: Spatial SU(2) Invariance Leaves the Temporal Coefficient Free.

Overview

An emergent Lorentzian co-metric on the product $\mathbb{R}_\tau\times\mathrm{Heis}_3(\mathbb{R})$ of an ordering line and a Heisenberg spatial factor carries a temporal coefficient $A_\tau$ and spatial coefficients. A four-dimensional spacetime needs their relative normalisation. Under [H-L] and [H-lift], Q5b gives the rank-three co-metric $\mathrm{diag}(-A_\tau, A_H, A_H, 0)$ and leaves the values of $A_\tau$ and $A_H$ open (open problem Q5b-O3). For a supplied full-rank extension invariant under the spin-one action, Q8 proves that the three spatial coefficients coincide, with common value $2\lambda$ and $\lambda > 0$ undetermined.

Q11 determines what invariance under a spatial $\mathrm{SU}(2)$ action can contribute to the temporal coefficient. The answer is structural: the ordering line carries the trivial representation, and Schur's lemma leaves one free scalar per isotypic component.

Result. Every invariant symmetric form on the cotangent fibre $\mathbb{R}_\tau\oplus W$ is $\alpha\,k_\tau^2 + \beta\,Q_{\mathrm{sp}}$ with $\alpha$ and $\beta$ independent. Invariance excludes mixed temporal–spatial terms but relates the two scalars by no equation: $A_H = 2\lambda$ gives no value of $A_\tau$, in particular not $A_\tau = 2$. Q11 closes a derivation route, not the possibility $A_\tau = 2$.

Quick facts

Author

Jérôme Beau

Work type

Preprint

Focus

Temporal normalisation, Schur's lemma, invariant quadratic forms, spin-one module, effective co-metric

Key result

Spatial $\mathrm{SU}(2)$ invariance leaves $A_\tau$ independent of the spatial coefficients; no coefficient value is derived

Status

Proved under the supplied action [ACT$_\tau$]; the co-metric with two independent scales is conditional on [H-L], [H-lift], [T] and [INV$^+$]; the coupling datum is open

Depends on

Q5b ([H-L], [H-lift]); Q7 ([ACT], supplied: no source supplies it); Q8 ([T], [INV$^+$])

Feeds into

Q12 only for its emergent-base interpretation under [H-L], [H-lift] and a supplied rank-four extension; the fixed-base Yang–Mills theorem is independent of Q11

Main results

Theorem 1.1 (proved). With $\mathrm{SU}(2)$ acting trivially on the ordering line and by the real spin-one representation on the spatial sector $W$ (supplied hypothesis [ACT$_\tau$]), every invariant symmetric bilinear form on $\mathbb{R}_\tau\oplus W$ is $\alpha\,k_\tau^2 + \beta\,Q_{\mathrm{sp}}$ with $\alpha$ and $\beta$ independent. There is no mixed temporal–spatial term, so the assumption of Q8 Corollary 6.1 is derived for an invariant symbol; fixing $A_H = 2\lambda$ imposes no condition on $A_\tau$.
Proposition 5.2 (proved). For every linear $\mathrm{SU}(2)$ action on a real four-dimensional space, the invariant forms are either the multiples of one definite form (the realified spin-$\tfrac12$ case) or a family of dimension at least two. No $\mathrm{SU}(2)$-invariance condition fixes a Lorentzian co-metric up to an overall factor.
Modelling identification 3.1 (modelling input). The ordering coordinate is $\tau = n\,\Delta\tau$, breadth-first depth times a free step $\Delta\tau$; the ordering derivative is defined relative to that step. This is a modelling input, not a lemma.
Proposition 4.1 (proved). Temporal rescalings commute with any action trivial on the ordering line and multiply $A_\tau$; for fixed discrete ordering data $A_\tau = a\,\Delta\tau^2$. No equivariance condition fixes the step.
Corollary 6.1 (conditional). Under [H-L] and [H-lift] of Q5b, [T] and [INV$^+$] of Q8, an invariant four-dimensional symbol and a negative ordering term, the co-metric is $\mathrm{diag}(-A_\tau, 2\lambda, 2\lambda, 2\lambda)$ with two independent scales. A choice of temporal unit turns it into $2\lambda\,\eta^{\mu\nu}$.
Remark 6.3 (status of Q5b-O3). The temporal half of Q5b-O3 amounts to one datum coupling the temporal and spatial normalisations, for instance the step $\Delta\tau$ expressed in spatial units. No source supplies it.

Scope. Q11 does not show that $A_\tau = 2$ is false. It shows that spatial $\mathrm{SU}(2)$ invariance does not derive it: the Casimir eigenvalue $2$ refers to one normalisation of the generators and determines no coefficient. Candidate sources of the coupling datum (a saturated capacity bound, the unit advance of the breadth-first front) are listed in the paper as exploratory proposals, not results.

Open problems

Q11-O1 (coupling datum). Supply, from the structure of the cascade, one datum coupling the temporal and spatial normalisations: the step $\Delta\tau$ in spatial units for fixed discrete ordering data, or directly the ratio $A_\tau/A_H$.
Q11-O2 (sign of the ordering term). Invariance fixes neither the ratio nor the sign of $\alpha$ relative to $\beta$; Q5b stipulates the Lorentzian sign. Derive it from the ordering structure.
Q11-O3 (front speed and symbol). Relate the unit advance of the breadth-first front in word length to the principal symbol of a limit operator, taking into account that the Carnot limit of the spatial factor is sub-Finsler; decide whether this yields the coupling datum or only a bound.

Interpretive outlook

Status: interpretation, not a result. In the reading in which $\tau$ is the parameter of projective ordering and the Heisenberg factor carries relational space, spatial isotropy and the unit of time are separate pieces of information. Spatial isotropy fixes the shape of the spatial form and forbids any preferred mixing of time with space, but the rate at which ordering advances relative to spatial relational distance, which a kinematic reading would call the invariant speed, is an independent datum that the representation theory of space alone does not supply.

Relation to the Cosmochrony programme

Q11 answers the temporal half of Q5b-O3 negatively along the representation-theoretic route and names what is missing. It establishes no value of $A_\tau$, of $A_H$, or of their ratio.

Q12: its vertical-variation theorem holds at an arbitrary supplied fixed base metric and does not depend on Q11. Only its interpretation on the Cosmochrony emergent base uses the Q11 co-metric under [H-L], [H-lift] and a supplied rank-four extension.
Lorentz-capacity papers: the Lorentz-capacity series, including the temporal-projection paper, takes the conditional form of Corollary 6.1 with two independent coefficients; it uses only its signature, decoupled temporal block and isometry group, and treats $g = 2\eta$ as a choice of units. Its kinematic chain names the identification of the light-cone speed with the Born–Infeld bound per ordering step as the hypothesis [H-cone], which is the coupling datum of Q11-O1 and is not derived. Q14 uses only the Lorentzian signature of Q11's conditional co-metric (Corollary 6.1) and states the spin soldering it needs as a hypothesis; Q11 supplies neither that soldering, nor coefficient values, nor an order generator.

References

Jérôme Beau. Toward a Temporal Scale for Emergent Spacetime: Spatial SU(2) Invariance Leaves the Temporal Coefficient Free, 2026. doi:10.5281/zenodo.20098387